Matrices of local dependence between spatial extreme events
Fonseca,C.∗, Martins, A.P., Pereira, L. and Ferreira, H. † November 10, 2009
Abstract: If a spatial process {Xt}t∈Z2 is isotropic then the usual pairwise
extremal dependence measures depend only on the distance k i − j k between the locations i and j.
In general, we need to evaluate the spatial dependence in each one of the eight directions of Z2. We shall consider matrices of multivariate tail and extremal coefficients. We table in matrices the degrees of dependence for chosen pairs of sets A and B of locations. The well known relation between the bivariate tail and extremal coefficients can be generalized in this multidirectional approach. The use of such measure-matrices is illustrated in a particular max-stable ran-dom field.
1
Introduction
In multivariate and spatial problems attention has often focused on obtaining dependence measures that capture the main characteristics of the dependence structure. For a max--stable stationary random field X = {Xt}t∈Z2 the extremal coefficient, ε(i, j) defined in
Schlather [4] and Schlather and Tawn [5] as
P (max (Xi, Xj) ≤ u) = Pε(i,j)(Xi≤ u) (1)
provides information about pairwise extremal dependence of X.
Unlike a Gaussian process, the dependence structure of a max-stable process is not com-pletely
characterized by pairwise dependence structure. Schlather and Tawn [5] extend the defi-nition of the extremal coefficient to a multivariate setting of any dimension, as follows
P max i∈A Xi ≤ u = Pε(A)(Xi ≤ u) , A ⊂ Z2. (2) ∗
Instituto Polit´ecnico da Guarda, Portugal, E-mail: cfonseca@ipg.pt
†
Departamento de Matem´atica, Universidade da Beira Interior, Portugal, E-mail: amartins@mat.ubi.pt, lpereira@mat.ubi.pt, helena@mat.ubi.pt
This coefficient measures the extremal dependence between the variables indexed by set A. The simple interpretation of ε(A) as the effective number of independent variables in the set A from which the maximum is drawn has led to its use as a dependence measure in a range of practical applications.
The extremal coefficient ε(A) is related to the upper tail dependence parameter defined in Joe ([2]) (see also Li [3] and Ferreira [1]) as
λ (A, B) = lim x→xFP \ i∈A {Xi > x} | \ i∈B {Xi> x} ! (3) where, A and B are disjoint regions of Z2, xF denotes the upper end point of F . This
parameter measures the tendency for large values of X at separate regions of locations to occur simultaneously. If in (3) we take A = {i} and B = {j} we obtain the bivariate tail dependence coefficient introduced by Sibuya [6]. In this case, if the probability in (3) is nonzero (zero), id est, the most extreme values can (can not) occur together, we say that the random field is pairwise asymptotically dependent (pairwise asymptotically independent) for locations i and j. The bivariate tail is related with the pairwise extremal coefficient through
λ (i, j) = 2 − ε (i, j) . (4)
If the spatial process X is isotropic then the pairwise extremal dependence measures depend only on the distance ki − jk between the locations i and j. In general we don’t have isotropy, so we need to evaluate the spatial dependence in each one of the eight directions of Z2. The aim of this paper is to define matrices of multivariate tail dependence, where each element measures the dependence for particular pair of sets A and B of locations. The use of such measure matrices is illustrated in particular max-stable random fields.
2
Characterizing local dependence in extreme events
For each i = (i1, i2) ∈ Z2, let sj(i), j = 1, 2, . . . , 8, be the neighbors of i defined as follows:
s1(i) = (i1+ 1, i2), s2(i) = i + 1, s3(i) = (i1, i2+ 1), s4(i) = (i1− 1, i2+ 1),
s5(i) = (i1− 1, i2), s6(i) = i − 1, s7(i) = (i1, i2− 1), s8(i) = (i1+ 1, i2− 1).
We define the boundary of A ⊆ Z2 by
Fr(A) = {i ∈ A : sj(i) /∈ A for some j ∈ {1, 2, . . . 8}}
and consider
Tsj(A) = {sj(i) : i ∈ Fr(A)} − A, j = 1, 2, . . . , 8,
as being the set containing the translations of the elements of F r(A) along the sj direction.
Our primary measure is a matrix, Λ Tk s(A), A
ele-Tskj(A) = {skj(i) : i ∈ F r(A)} − A, k ≥ 2, where skj denotes sj◦ . . . ◦ sj, k times, to occur simultaneously. ΛTsk(A), A= λ Tsk4(A), A λ Tsk3(A), A λ Tsk2(A), A λ Tk s5(A), A 1 λ Tk s1(A), A λ Tsk6(A), A λ Tsk7(A), A λ Tsk8(A), A (5)
where 1 = λ (A, A) = λ (A, Ts0(A)) and Ts0(A) = A.
In what follows we shall denote the set {Xi : i ∈ A}, A ⊆ Z2, by XA and the vector
of dimension |A| with all components equal x by xA. The next result presents a recursive
formula to obtain the matrices Λ Tk
s(A), A, k ∈ N. Proposition 2.1 If αTskj(A)= lim x→xF 1 − PXTk sj(A)≯ xTsjk(A) 1 − PXTk sj(A)≤ xTsjk(A) , k ∈ N, α(A) = lim x→xF 1 − P (XA≯ xA) 1 − P (XA≤ xA) , α(A) > 0 and β(Tskj(A), A) = lim x→xF 1 − PXTk sj(A)≯ xTsjk(A), XA ≯ xA 1 − PXTk sj(A) ≤ xA, XA≤ xA , k ∈ N
then the coefficients λ(Tskj(A), A), k ≥ 2, satisfy λ(Tskj(A), A) = λ(Tsk−1j (A), A) −α(T k−1 sj (A)) α(A) × ε(Tsjk−1(A)) ε(A) + +β(T k−1 sj (A),A) α(A) × ε(Tsjk−1(A)∪A) ε(A) + +α(T k sj(A)) α(A) × ε(Tk sj(A)) ε(A) − −β(T k sj(A),A) α(A) × ε(Tk sj(A)∪A) ε(A) .
λ(Tskj(A), A) = lim
x→xF
P
XTk
sj(A)> xA, XTsjk−1(A)> xTsjk−1(A), XA > xA
1 − P (XA ≯ xA) + + lim x→xF P XTk
sj(A) > xA, XTsjk−1(A) ≯ xTsjk−1(A), XA> xA
1 − P (XA≯ xA) = = λ(Tsk−1j (A), A) − lim x→xF 1 − P XTk−1 sj (A)≯ xTsjk−1(A) 1 − P (XA ≯ xA) + + lim x→xF 1 − P XTk−1 sj (A) ≯ xTsjk−1(A), XA ≯ xA 1 − P (XA≯ xA) + + lim x→xF 1 − PXTk sj(A) ≯ xTsjk(A) 1 − P (XA ≯ xA) − − lim x→xF 1 − P (XTk sj(A) ≯ xTsjk(A), XA≯ xA) 1 − P (XA≯ xA) = = λ(Tsk−1j (A), A) − α(T k−1 sj (A)) α(A) × limx→xF 1 − P XTk−1 sj (A) ≤ xTsjk−1(A) 1 − P (XA ≤ xA) + +β(T k−1 sj (A),A) α(A) × limx→xF 1 − P XTk−1 sj (A)≤ xTsjk−1(A), XA ≤ xA 1 − P (XA≤ xA) + +α(T k sj(A)) α(A) × limx→xF 1 − PXTk sj(A) ≤ xTsjk(A) 1 − P (XA ≤ xA) − −β(T k sj(A),A) α(A) × limx→xF 1 − PXTk sj(A)≤ xTsjk(A), XA ≤ xA 1 − P (XA ≤ xA) . From (2) we obtain the following relation
λ(Tk sj(A), A) = λ(T k−1 sj (A), A) − α(Tsjk−1(A)) α(A) × ε(Tsjk−1(A)) ε(A) + +β(T k−1 sj (A),A) α(A) × ε(Tsjk−1(A)∪A) ε(A) + +α(T k sj(A)) α(A) × ε(Tk sj(A)) ε(A) − −β(T k sj(A),A) α(A) × ε(Tk sj(A)∪A) ε(A) , k ≥ 2.
where 2 denote the square matrix with all elements equal 2 and
ε({i} , Tsk({i}) =
ε({i} , Tsk4({i}) ε({i} , T
k
s3({i}) ε({i} , T
k s2({i})
ε({i} , Tsk5({i}) 1 ε({i} , T
k s1({i}) ε({i} , Tk s6({i}) ε({i} , T k s7({i}) ε({i} , T k s8({i}) . (6)
3
Examples
We will now illustrate our measure matrices of dependence with a dependent random field. Let Y = {Yt}t∈Z2 be an i.i.d random field with unit Fr´echet margins. From Y we define
a stationary and anisotropic random field X = {Xt}t∈Z2 as follows
Xi= max{Ysj(i), j ∈ {1, . . . , 8}}. 0 5 10 15 20 0 5 10 15 20 0.5 1 i j X(i ,j ) 2 4 6 8 10 12 14 16 18 20 5 10 15 20
Figure 1: Simulation of the random field X
The matrices of bivariate tail dependence coefficients, until achieved independence in all directions, are
Λ(T1 s({i}), {i}) = 2 4 2/8 4/8 2/8 4/8 1 4/8 2/8 4/8 2/8 3 5 Λ(T 2 s({i}), {i}) = 2 4 1/8 3/8 1/8 3/8 1 3/8 1/8 3/8 1/8 3 5 Λ(T 3 s({i}), {i}) = 2 4 0 0 0 0 1 0 0 0 0 3 5
With k = 3 we get the independence in all directions. Each one of the last matrices measures the tail dependence between {i} and its neighbors in some step. We illustrate the values of these matrices in figure 2. In that, the light color means less dependence and the dark more dependence, then black is total dependence and white is independence.
The matrices of bivariate extremal coefficients for this random field are
ε({i}, T1 s({i})) = 2 4 14/8 12/8 14/8 12/8 1 12/8 14/8 12/8 14/8 3 5 ε({i}, T 2 s({i})) = 2 4 15/8 13/8 15/8 13/8 1 13/8 15/8 13/8 15/8 3 5 ε({i}, T 3 s({i})) = 2 4 2 2 2 2 1 2 2 2 2 3 5
The measure of the tendency for large values of X at the regions A = {i, i + 1} ant its translation along each direction, to occur simultaneously, is given by
3 3 3 2 2 2 1 1 1 3 2 1 1 2 3 1 1 1 2 2 2 3 3 3
Figure 2: Representation of matrices values of bivariate tail dependence coefficients for Y at steps 1, 2 and 3. Λ(Ts1(A), A) = 1/2 0 0 0 1 0 0 0 1/2 ,
and the independence in all directions was achieved in two steps,
Λ(Ts2(A), A) = 0 0 0 0 1 0 0 0 0 . 2 2 2 1 1 1 2 1 1 2 1 1 1 2 2 2
Figure 3: Representation of matrices values of the tail dependence coefficients for the sets of locations A = {i, i + 1} and Tsk(A), k = 1, 2.
References
[1] H. Ferreira. Measuring dependence of two multivariate extremes. Notas e Comu-nica¸c˜oes do CEAUL 08/08, 2008.
[2] H. Joe. Parametric family of multivariate distributions with given margins. J. Multi-variate Anal., (46):262–282, 1993.
[3] H. Li. Orthant tail dependence of multivariate extreme value distributions. Journal of Multivariate Analysis, (100):243–256, 2009.
[4] M. Schlather. Models for stationary max-stable random fields. Extremes, (5(1)):33–44, 2002.
[5] M. Schlather and J. Tawn. A dependence measure for multivariate and spatial ex-tremes: Properties and inference. Biometrika, (90(1)):139–156, 2003.
